Prove the equality (1 + i*3^1/2 )^2 + (1 - i*3^1/2 )^2 = -4

We'll expand the squares from the left sides, using the
formula:

(a+b)^2 = a^2 + 2ab +
b^2

We'll put a = 1 and b =
i*sqrt3

(1 + i*sqrt3 )^2 = 1 + 2i*sqrt3 +
3i^2

But i^2 = -1

(1 + i*sqrt3
)^2 = 1 + 2i*sqrt3 - 3

We'll combine like
terms:

(1 + i*sqrt3 )^2 = 2i*sqrt3 - 2
(1)

Now, we'll expand the square (1 - i*sqrt3
)^2:

(1 - i*sqrt3 )^2 = 1 - 2i*sqrt3 -
3

(1 - i*sqrt3 )^2 = -2i*sqrt3 - 2
(2)

We'll add (1) and
(2):

2i*sqrt3 - 2 - 2i*sqrt3 - 2 =
-4

We'll eliminate like
terms:

-4 =
-4

The identity is
verified!